FusionAtlasPaths={"~/projects/fusionatlas/code/package/","~/Documents/research/fusionatlas/package/","C:\\Documents and Settings\\James Tener\\My Documents\\Research\\fusionatlas\\package\\"};
$Path=$Path~Join~FusionAtlasPaths;
<<FusionAtlas`

BadSeed={BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1,2},{2,1}]]}

GraphToString[BadSeed[[1]]]

GraphToString[BadSeed[[2]]]

N[DimensionOfGenerator[BadSeed[[1,1]]]]

FindBigraphExtensions[5][BadSeed[[1,1]],3]

Map[DisplayBigraph,BadSeed]

dim=N[qFromd[DimensionOfGenerator[BadSeed[[1,1]]]]]

NSolve[q^3-q^2-1==0]

qI[n_,q_]:=Together[(q^(n)-q^(-n))/(q-q^(-1))]

qI[n,q]

The consistency is obtained from the \ell^2 eigenvector equation.

Consistency=Factor[Together[qI[2,q]qI[n+2,q]/2-q^2 qI[n+1,q]/(q^2-1)]]

ToString[Consistency,TeXForm]

Plot[Consistency/.(n-> 3),{q,1.543,1.545}]

Connection=Factor[Together[qI[2n+7,q]-qI[2n+5,q]-qI[2n+3,q]+qI[2n+1,q]-4qI[n+1,q]^2]]

ToString[Connection,TeXForm]

Plot[Connection/.(n-> 3),{q,1.543,1.545}]

The "consistency" appears as a factor of the "connection." It is clear that "consistency"=1+(q^4)-(q^(4+2 n))-2 (q^(6+2 n))+(q^(8+2 n))  < 1-2 (q^2)-(q^4)+(q^(4+2 n))+(q^(8+2 n)),  so the biggest root of the "consistency" is the biggest root of the "connection".

NSolve[(Consistency/.(n-> 3))==0,q];

NSolve[(Connection/.(n-> 3))==0,q];

NSolve[(Consistency/.(n-> 4))==0,q];

NSolve[(Connection/.(n-> 4))==0,q];

Find some bounds on  1-2 (q^2)-(q^4)+(q^(4+2 n))+(q^(8+2 n)) so we can rule out everything but graphs with infinite legs: 
For n ≥ 3, 
1-2 (q^2)-(q^4)+(q^(4+2 n))+(q^(8+2 n)) ≥ 1-2 (q^2)-(q^4)+(q^10)+(q^14). Clearly this has a zero at q=1. Taking derivatives, we get

D[(foo/.(n->3)),q]

which is clearly positive for all q≥1. Hence   1-2 (q^2)-(q^4)+(q^(4+2 n))+(q^(8+2 n)) is always strictly positive for q>1.

Plot[(1+(q^4)-(q^(4+2 n))-2 (q^(6+2 n))+(q^(8+2 n)))/.(n-> 6),{q,1,1.6}]

Estimates for Lemma A.7:

Together[1+2/(q^(2k))-(q+q^(-1))^2(1/(q^(2))+1/(2q^(2k+2)))]

Negative! this is bad news...

estimate=Together[1+2/(q^(2k))-(q+q^(-1))^2(1/(q^(2k-2))+1/(2q^(2k+2)))]

Plot[estimate/.(k-> 1),{q,1,1.8}]

Plot[estimate/.(k-> 2),{q,1,1.8}]

Plot[estimate/.(k-> 3),{q,1,1.8}]

NSolve[(estimate/.(k->3))==0]

Plot[estimate/.(k-> 4),{q,1,1.8}]

estimate2=Together[1/(q^4)+2/(q^(2k))-(q+q^(-1))^2(1/(q^(6))+1/(2q^(2k+2)))]

Plot[estimate2/.(k-> 6),{q,1,1.8}]

NSolve[(estimate2/.(k-> 4))==0]

AddTriple[g_GradedBigraph,m_,k_,n_]:=GradedBigraph @@ (Table[{{1}},{i,1,m}]~Join~(List @@ g)~Join~Table[{{1,0},{0,1}},{i,1,k}]~Join~Table[{{1}},{i,1,n-k}]~Join~{{{1},{1}}})

AddTriple[BadSeed[[1,1]],2,2,2]

DisplayBigraph[foo]

foo=GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{1,0}}]

N[DimensionOfGenerator[foo],50]

(q+q^(-1))/.NSolve[(1+(q^4)-(q^(4+2 n))-2 (q^(6+2 n))+(q^(8+2 n))/.(n-> 3))==0,q]

2.191520911042651`

qFromd[2.191520911042651`]

We see that the norm of the graph pictured above is smaller than the norm of \Gamma_{3,\infty}.

AddTriple2[g_GradedBigraph,m_,n_]:=GradedBigraph @@ (Table[{{1}},{i,1,m}]~Join~(List @@ g)~Join~Table[{{1,0},{0,1}},{i,1,n}]~Join~{{{1,0},{1,0},{0,1}}})

DisplayBigraph[AddTriple2[BadSeed[[1,1]],0,12]]

N[DimensionOfGenerator[AddTriple2[BadSeed[[1,1]],0,12]]]

2.1915209330290684`

qFromd[N[DimensionOfGenerator[AddTriple2[BadSeed[[1,1]],0,8]]]]

Plot[2^(1/2)q-(q+q^(-1)),{q,1.53,1.6}]

Plot[1+3q^2-3(q+q^(-1))^2,{q,1.53,1.6}]

a=qI[2,q]qI[n+2,q]/(2q^3)/q^k

PositiveInequality=Together[a q^4+(a^2+a)q^2-6a^2q^2-3a^2]

Plot[(PositiveInequality/.(n->3))/.k->1,{q,1.53,1.6}]

NSolve[2q^2-3q^(-2)-3==0,q]

Solve[2q^2-3q^(-2)-3==0,q]

Factor[q^4+2d q+d^2q^2+2q^2+1-d^2(2q^2+2)/.{d->(q+q^(-1))}]

Factor[2(d^2q)^2+(d q+1)^2+d^2+(d q+d q^(1)+q^(-2))^2+(d+2/q)^2+2(d/q+q^(-2))^2-d^4(2q^2+2+2/q^2+2/q^4)/.{d->(q+q^(-1))}]

Plot[-2-6 (q^2)+(q^4)-2 (q^6)+3 (q^8),{q,1,1.6}]

NSolve[-2-6 (q^2)+(q^4)-2 (q^6)+3 (q^8)]

N[qFromd[DimensionOfGenerator[BadSeed[[1,1]]]]]

N[((2(d^2q)^2+(d q+1)^2+d^2+(d q+d q^(1)+q^(-2))^2+(d+2/q)^2+2(d/q+q^(-2))^2)/(d^4(2q^2+2+2/q^2+2/q^4))/.{d->(q+q^(-1))})/.{q->1.539222338420433}]

NSolve[2q^2-3-3q^-2]

foo=GraphFromString["(bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0v1x1duals1v1v1x2v1x2v2x1v1, bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1v0x1p0x1p1x0p1x0duals1v1v1x2v1x3x2x4v1x2v4x3x2x1)"]

DisplayBigraph/@foo

bar=GraphFromString["(bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1v1x1duals1v1v1x2v1x3x2x4v1x2v1, bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0v0x1p0x1p1x0p1x0duals1v1v1x2v1x2v2x1v2x1x4x3)"]

DisplayBigraph/@bar

baz = GraphFromString["(bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0v0x1p1x0duals1v1v1x2v1x2v2x1v1x2, bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1v0x1p1x0duals1v1v1x2v1x3x2x4v1x2v2x1)"]

DisplayBigraph/@baz

DisplayBigraph/@BadSeed

OcneanuDisplay[BadSeed[[1]],BadSeed[[2]]]

OcneanuDisplay[baz[[1]],baz[[2]]]